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on extracting band-pass signal by using Goertzel algorithm instead of DFT filter banks

Started by Sun Lei. June 28, 2007
 On the poly-phase filter banks, it's straightforward to use the
Goertzel agorithm instead of DFT can reduce the costs if only one user
is to be extracted. In the Tim Hentschel and Tuttlebee's book and some
related articles, there're some discussions on this topic. The
following link is available, related discussions on p.p. 31.
    http://citeseer.ist.psu.edu/hentschel02channelization.html
    While I check this algorithm, I found a question. The goertzel
algorithm is a single-pole IIR filter, which means that this filter
has a very narrow frequncy response at the pole bin. This can be
easily simulated and confirmed by MATLAB. So, when I use the goertzel
algorithm to exract a band-limited channel, the result is that, there
is a 'pole frequency response' at the corresponding pole of goertzel
filter. And the extracted user band is heavily distorted. This can be
seen on the picture follows.The pciture is to extract a chirp signal
from Band 0 of a 8-equal distributed band within a wideband(original
intention is to simulate a uniform DFT filter banks). The polyphase
branches and the decimation number is M=8. In the goertzel on the
figure, k =0.1, N =8. on  exp(-j*2*pi*k/N). The pole distortion is at
point ''A'' on the figure.

         |
         |
         |
         |
         | --------.                          | ------|
         |         '\                        .'       |
         |          \.                       |        |
         |           \                      |'        |
         |           '.                     |         |
         |            \                    |          |
         |             \                   |          |
         |             '.                 |           |
         |              \                .'           |
         |                ---------------'            |
          --------------------------------------------|-
        o                                              1
                  Original Chirp Signal Spectrum

         |   ||
         |   ||
         |   ||
         |  .||
         | --|||---.                          | ------|
         |   A     '\                        .'       |
         |          \.                       |        |
         |           \                      |'        |
         |           '.                     |         |
         |            \                    |          |
         |             \                   |          |
         |             '.                 |           |
         |              \                .'           |
         |                ---------------'            |
          --------------------------------------------|-
        o                                              1

        Band signal extracted by using Goertzel algorithm
(created by AACircuit v1.28.6 beta 04/19/05 www.tech-chat.de)


    So, I think, the goertzel algorithm can be used to extract a
single-tone on channelization, not suitable for a band-limited user.

    But, On the channelization theory, the goertzel algorithm applied
here is just an operation, it's to substitute the frequency
modulations on H(K), so, it should has similar effects compare with
DFT operations.While the result is not.This question confused me a
long time, I wonder if you smart guys can give me some clues.


Sun Lei.